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**A linear expected-time algorithm for computing planar relative neighbourhood graphs.**
*(English)*
Zbl 0653.68034

A new algorithm for computing the relative neighbourhood graph (RNG) of a planar point set is given. The expected running time of the algorithm is linear for a point set in a unit square when the points have been generated by a homogeneous planar Poisson point process. The worst-case running time is quadratic on the number of the points. The algorithm proceeds in two steps. First, a supergraph of the RNG is constructed with the aid of a cell organization of the points. Here, a point is connected by an edge to some of its nearest neighbours in eight regions around the point. The nearest region neighbours are chosen in a special way to minimize the costs. Second, extra edges are pruned from the graph by a simple scan.

### MSC:

68Q25 | Analysis of algorithms and problem complexity |

68R10 | Graph theory (including graph drawing) in computer science |

52A37 | Other problems of combinatorial convexity |

### Keywords:

Voronoi diagram; cell technique; region approach; computational geometry; relative neighbourhood graph; planar point set
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\textit{J. Katajainen} et al., Inf. Process. Lett. 25, 77--86 (1987; Zbl 0653.68034)

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### References:

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